In this thesis we describe the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds, and develop suitable quantization techniques. For this, we propose a Courant sigma-model on an open membrane with target space M, which we regard as a topological sector of closed string dynamics on Rspace. We then reduce it to a twisted Poisson sigma-model on the boundary of the membrane with target space the cotangent bundle T M. The pertinent twisted Poisson structure is provided by a U(1) gerbe in momentum space, which geometrizes R-space. From the membrane perspective, the path integral over multivalued closed string fields in Q-space (i.e. the T-fold endowed with a non-geometric Q- flux which is T-dual to the R-flux), is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's global deformation quantization formula for the twisted Poisson manifolds, which we take as our proposal for quantization. We calculate the corresponding nonassociative star product and its associator, and derive closed formulas for the case of a constant R-flux. We then develop various versions of the Seiberg{Witten map, which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We also propose a second quantization method based on quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group. This formalism provides a categori cation of Weyl's quantization map, and leads to a consistent quantization of Nambu{Poisson 3-brackets. We show that the convolution product coincides with the star product obtained by Kontsevich's formula, and clarify its relation with the twisted convolution products for topological nonassociative torus bundles. As a first step towards formulating quantum gravity on non-geometric spaces, we develop a third quantization method to study nonassociative deformations of geometry in R-space, which is analogous to noncommutative deformations of geometry (i.e. noncommutative gravity). We find that the symmetries underlying these nonassociative deformations generate the non-abelian Lie algebra of translations and Bopp shifts in phase space. Using a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions, and the exterior differential calculus in R-space. We define a suitable integration on these nonassociative spaces, and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. In this setting, we consider extensions to non-constant R-flux backgrounds as well as more generic twisted Poisson structures emerging from non-parabolic monodromies of closed strings. As a first application of our nonassociative star product quantization, we develop nonassociative quantum mechanics based on phase space state functions, wherein 3-cyclicity is instrumental for proving consistency of the formalism. We calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux. For a second application, we construct nonassociative deformations of fields, and study perturbative nonassociative scalar field theories on R-space. We nd that nonassociativity induces modi cations to the usual classi cation of Feynman diagrams into planar and non-planar graphs, which are controlled by 3-cyclicity. The example of '4 theory is studied in detail and the one-loop contributions to the two-point function are calculated.